Solve the inequality and graph the solution.

s
3
≥3
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of the segment, ray, or line to delete it.

1 answer

To solve the inequality \( s^3 \geq 3 \), we start by isolating \( s \).

  1. Take the cube root of both sides: \[ s \geq \sqrt[3]{3} \]

    Since the cube root function is increasing, there are no sign changes needed when taking the cube root.

  2. Identify the solution: The solution to the inequality is \( s \) such that: \[ s \geq \sqrt[3]{3} \]

  3. Graph the solution:

    • The endpoint \( s = \sqrt[3]{3} \) is included in the solution, so we will use a closed dot.
    • The solution extends to positive infinity.
    • We represent this graphically as a line starting at \( \sqrt[3]{3} \) (closed dot) and extending to the right.

Here’s a description of the graphing actions:

  • Plot \( \sqrt[3]{3} \) on the number line with a closed endpoint (●).
  • Draw a line extending to the right of this point indicating all values greater than \( \sqrt[3]{3} \).

If you want to change the endpoint from closed to open for any reason, you could consider what happens if the inequality were strict (\( s^3 > 3 \)), but since here we are working with \( s^3 \geq 3 \), we keep it closed as stated above.

Thus, the final solution can be represented on a number line as follows:

\[ \text{Graph:} \]

<---|--------|--------|--------|--------|--------|--------|--------|--->
    -2      -1       0        1        2      sqrt[3]{3}       3      4
                        ●====================>

Here, the filled dot at \( \sqrt[3]{3} \) indicates that \( \sqrt[3]{3} \) is included in the solution, and the right-pointing arrow indicates that the solution goes to positive infinity.